Particle physicists have spent much of this century
grappling with one basic question in various forms: what are the
fundamental degrees of freedom needed to describe nature, and what are
the laws that govern their dynamics? First molecules, then atoms, then
“elementary particles” such as protons and neutrons all have been
revealed to be composite objects whose constituents could be studied as
more fundamental degrees of freedom. The current “standard model”
of particle physics—which is nearly 25 years old, has much
experimental evidence in its favor and is comprised of six quarks, six
leptons, four forces, and the as yet unobserved Higgs boson—contains
internal indications that it, too, may be just another step along the
path toward uncovering the truly fundamental degrees of freedom. The
standard model is valid to distances as small as
10−16 cm, and there is
some evidence (such as that obtained by extrapolating the strengths of
the four forces to determine the distance scale at which they might
become indistinguishable) that the next level of structure will be
detected only at a distance scale of roughly
10−32 cm, far beyond our
abilities to measure in the laboratory.

The study of motion and gravity also has undergone several revisions
during this century. Reconciling the Newtonian theory of motion with
the experimentally observed constancy of the speed of light required
the introduction of special relativity, which quite remarkably insists
that space and time are intimately related, much as different faces of
the same coin. Incorporating gravity into this framework required an
even more drastic modification of our view of space and time: in
general relativity, space time is seen as intrinsically warped, and the
warping is responsible for the gravitational force. General relativity
is a “classical” theory, which takes no notice of effects from
quantum mechanics (whose development was another of the triumphs of
theoretical and experimental physics in the early part of the century).
A serious problem arises when general relativity is extrapolated to
tiny distance scales (again roughly
10−32 cm) where quantum
effects must be taken into account: the quantum-mechanical perturbation
expansion of this theory has uncontrollable divergences. Lessons
learned from the history of particle physics suggest that this should
be a signal of new physics occurring at these tiny distance scales.

String theory offers a hope of addressing both of these issues. There
is only one known way to “smear out” the gravitational
interaction and hence cure the divergence problem in the
quantum-mechanical expansion of general relativity: model the particles
in the theory not as points, but as one-dimensional loops of
“string.” In fact, every consistent such string model necessarily
contains a special kind of particle—the graviton—whose long-distance
interactions are described by general relativity. So in a sense, string
theory predicts gravity. Moreover, some of the simplest string
theories, the Calabi–Yau models, closely resemble unified
(super-symmetric) versions of the standard model.

However, before recent developments our understanding of string theory
was limited to situations in which small numbers of strings interact
weakly. This is unsatisfactory for several reasons. First, we are
undoubtedly missing important dynamical effects in such a limited
study. (Analogous effects in quantum field theory such as quark
confinement and spontaneous symmetry breaking are crucial ingredients
in the standard model.) And second, we are really lacking the
fundamental principle underlying string theory. It is quite likely that
the theory will look very different once this principle is uncovered.

An exciting new frontier was opened during the past few years with the
discovery of “string duality,” which predicts equivalences among
various seemingly different physical systems. This discovery has its
roots in the properties of super-symmetry, a novel type of symmetry
that all consistent string theories possess. Briefly, super-symmetry
relates properties of two basic types of particles—bosons and
fermions—which cannot be related by any ordinary symmetry. There are a
number of good reasons for suspecting that super-symmetry will play a
role in the structure of particle physics beyond the standard model.
And it turns out that super-symmetric theories are highly constrained,
and certain properties of them can be identified, which can be
calculated when the interactions are weak, yet cannot change when the
interactions become small. That invariance under variation
of the interaction strength was key to the discovery of string
duality.

One of the important achievements of string duality has been the
determination of the behavior of all of the various consistent string
theories (there are five of them) when the interactions become strong.
Surprisingly, they all are related to each other, and to one additional
theory—not quite a string theory—known as M-theory. The duality
relationships introduce additional objects into the theory known as
D-branes, which may have 0, 1, 2, 3, … spatial dimensions. (This
answers one natural objection to the introduction of strings in the
first place: why stop at one-dimensional objects?) The relationships
among these theories are illustrated in
Fig. 1, which shows six limit
points in a large parameter space that admit six different descriptions
as a string theory or M-theory. We do not yet understand how to
describe the theory when the parameters are out in the middle of this
space—the sought-after fundamental principle underlying string theory
should provide such a description in the future.

Space of string vacua. The cusps are limits in
which a weakly coupled string description is possible, except for the
M-theory limit. [Reproduced with permission from Polchinski, J. (1996)
Rev. Mod. Phys. 68, 1245–1258; copyright 1996 by
the American Physical Society.]

A second important achievement of string duality has been yet another
drastic modification of our notion of space and time. The five
consistent string theories all involve 10 space-time dimensions, and
M-theory needs 11 space-time dimensions. All but four of these must be
curled up into a very tiny compact shape to produce a model of our
world, i.e., one with only four observable space-time dimensions. This
is an old idea, dating from the earliest attempts to unify gravity with
the other forces, and in such models detailed properties of the
elementary particles will be determined by the structure of the tiny
compact space. The quandary for string theory has been that there is a
very large number of such compact spaces to choose among. However, by
using string duality we have learned that when nonperturbative effects
are included, string theory manages to link together the models based
on many such compact spaces, effectively ripping and tearing the fabric
of space time yet in a perfectly smooth way using quantum effects. In
technical language, the topology of the universe itself is changed.
This is an exciting result, as it gives rise to renewed hope that there
may be only one possible string-theoretic model of the universe, and it
may be possible to eventually predict such features as particle masses
and interaction strengths directly from such a theory.

Development has been rapid on many fronts since string duality was
introduced. We may be seeing glimpses of the underlying principle
manifested in these new results. The challenging task that lies ahead
is to discover that principle, and thereby find what may well be the
truly fundamental degrees of freedom in our universe.

Footnotes

↵‡ To whom reprint requests should be addressed. e-mail:
drm{at}math.duke.edu.

This paper is a summary of a session presented at the Ninth
Annual Frontiers of Science Symposium, held November 7–9, 1997, at the
Arnold and Mabel Beckman Center of the National Academies of Sciences
and Engineering in Irvine, CA.